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A Mechanization of Strong Kleene Logic for Partial Functions
 PROCEEDINGS OF THE 12TH CADE
, 1994
"... Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. ..."
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Cited by 30 (11 self)
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Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.
Elimination of Cuts in Firstorder Finitevalued Logics
 J. Inform. Process. Cybernet. (EIK
, 1994
"... A uniform construction for sequent calculi for finitevalued firstorder logics with distribution quantifiers is exhibited. Completeness, cutelimination and midsequent theorems are established. As an application, an analog of Herbrand's theorem for the fourvalued knowledgerepresentation logic ..."
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Cited by 19 (4 self)
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A uniform construction for sequent calculi for finitevalued firstorder logics with distribution quantifiers is exhibited. Completeness, cutelimination and midsequent theorems are established. As an application, an analog of Herbrand's theorem for the fourvalued knowledgerepresentation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information.
Multipleconclusion LP and default classicality
 Review of Symbolic Logic
, 2011
"... Abstract. Philosophical applications of familiar paracomplete and paraconsistent logics often rely on an idea of ‘default classicality’. With respect to the paraconsistent logic LP (ithe dual of Strong Kleene or K3), such ‘default classicality ’ is standardly cashed out via an LPbased nonmonotoni ..."
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Cited by 11 (8 self)
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Abstract. Philosophical applications of familiar paracomplete and paraconsistent logics often rely on an idea of ‘default classicality’. With respect to the paraconsistent logic LP (ithe dual of Strong Kleene or K3), such ‘default classicality ’ is standardly cashed out via an LPbased nonmonotonic logic due to Priest (1991, 2006a). In this paper, I offer an alternative approach via a monotonic, multipleconclusion version of LP. §1. Introduction. The logic LP is the dual of the wellknown logic K3 (viz., Strong Kleene).1 This logic, like K3, has found prominent applications in philosophy, particularly with respect to paradoxical phenomena (Beall, 2009; Brady, 2006; Field, 2008; Kripke, 1975; Priest, 2006a,b). In such applications, the background picture is one of ‘default classicality’. The basic thought is that classical logic is
Labeled Calculi and Finitevalued Logics
, 1998
"... A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finitevalued logic if the labels are interpreted as sets of truth values (setsassigns). Furthermore, it is shown that any finitevalued ..."
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Cited by 7 (2 self)
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A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finitevalued logic if the labels are interpreted as sets of truth values (setsassigns). Furthermore, it is shown that any finitevalued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic in the number of truth values, and it is shown that this bound is tight. Keywords: finitevalued logic, labeled calculus, signed formula, setsassigns
Proof Theory of Finitevalued Logics
, 1993
"... Manyvalued logic is not much younger than the whole field of symbolic logic. It was introduced in the early twenties of this century by ̷Lukasiewicz [1920] and Post [1921] and has since developed into a very large area of research. Most of the early work done has concentrated on problems of axiomat ..."
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Cited by 6 (1 self)
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Manyvalued logic is not much younger than the whole field of symbolic logic. It was introduced in the early twenties of this century by ̷Lukasiewicz [1920] and Post [1921] and has since developed into a very large area of research. Most of the early work done has concentrated on problems of axiomatizability on the one hand, and algebraical/model theoretic investigations on the other. The proof theory of manyvalued systems has not been investigated to any comparable extent. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. Several people have, since the 1950’s, proposed ways to generalize such formalisms from the classical to the manyvalued case. One particular method for systematically obtaining calculi for all finitevalued logics was invented independently by several researchers, with slight variations in design and presentation. (Section 3.1 contains a short overview of work done in this area). The main aim of this report is to develop the proof theory of finitevalued first order logics in a
A Tableau Calculus for Partial Functions
, 1996
"... Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a threevalued logic decades ago, but there has not been a satisfactory mechanization. R ..."
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Cited by 6 (5 self)
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Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a threevalued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a tableau calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.
Conservatively extending classical logic with transparent truth
 Review of Symbolic Logic
, 2012
"... Abstract. This paper shows how to conservatively extend a classical logic with a transparent truth predicate, in the face of the paradoxes that arise as a consequence. All classical inferences are preserved, and indeed extended to the full (truthinvolving) vocabulary. However, not all classical met ..."
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Cited by 5 (2 self)
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Abstract. This paper shows how to conservatively extend a classical logic with a transparent truth predicate, in the face of the paradoxes that arise as a consequence. All classical inferences are preserved, and indeed extended to the full (truthinvolving) vocabulary. However, not all classical metainferences are preserved; in particular, the resulting logical system is nontransitive. Some limits on this nontransitivity are adumbrated, and two proof systems are presented and shown to be sound and complete. (One proof system features admissible Cut, but the other does not.) §1. Introduction. Adding a truth predicate to a language governed by classical logic is not easy. It is particularly tricky when the truth predicate is intended to be transparent— such that T 〈A 〉 and A are everywhere intersubstitutable. The trouble, as is wellknown, comes from such paradoxes as the liar; because of them, theories of truth typically either use a nontransparent truth predicate (Halbach, 2011; Maudlin, 2004) or a logic weaker than
Canonical Signed Calculi, Nondeterministic Matrices and Cutelimination, forthcoming
 in the Proceedings of LFCS 2009, LNCS
, 2009
"... Abstract. Canonical propositional Gentzentype calculi are a natural class of systems which in addition to the standard axioms and structural rules have only logical rules where exactly one occurrence of a connective is introduced and no other connective is mentioned. Cutelimination in such systems ..."
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Abstract. Canonical propositional Gentzentype calculi are a natural class of systems which in addition to the standard axioms and structural rules have only logical rules where exactly one occurrence of a connective is introduced and no other connective is mentioned. Cutelimination in such systems is fully characterized by a syntactic constructive criterion of coherence. In this paper we extend the theory of canonical systems to the considerably more general class of signed calculi. We show that the extended criterion of coherence fully characterizes only analytic cutelimination in such calculi, while for characterizing strong and standard cutelimination a stronger criterion of density is required. Modular semantics based on nondeterministic matrices are provided for every coherent canonical signed calculus. 1